Question

Consider the surface x 2 − 2 y 2 − 8 z 2 = 16. Which of the following is/are correct?

P. The traces parallel to the xy-plane are ellipses.
Q. The traces parallel to the xz-plane are hyperbolas.
R. The surface is a hyperboloid of two sheets.
A.Q and R
B.P and Q
C.P and R
D.P only
E.Q only

Answer 1

The traces parallel to the xy-plane are ellipses, the traces parallel to the xz-plane are hyperbolas, and the surface is a hyperboloid of two sheets.

Step-by-step explanation:

The equation x2 - 2y2 - 8z2 = 16 represents an ellipsoid in three-dimensional space.

The traces parallel to the xy-plane occur when z = k, where k is a constant. Substituting z = k into the equation gives x2 - 2y2 - 8k2 = 16, which is the equation of an ellipse. Therefore, option P is correct.

The traces parallel to the xz-plane occur when y = k, where k is a constant. Substituting y = k into the equation gives x2 - 8z2 = 16 - 2k2, which is the equation of a hyperbola. Therefore, option Q is correct.

The equation represents a surface with different cross-sections in different planes, indicating that the surface is a hyperboloid of two sheets. Therefore, option R is correct.